Certainly! Let's work through the problem step-by-step.
Step 1: Determine the slope of the original line
The equation of the given line is:
[tex]\[ y = -\frac{1}{2} x - 5 \][/tex]
From this equation, we can see that the slope (m) is:
[tex]\[ m = -\frac{1}{2} \][/tex]
Step 2: Find the slope of the perpendicular line
The slope of a line perpendicular to another line is the negative reciprocal of the original slope. So, we take the negative reciprocal of [tex]\(-\frac{1}{2}\)[/tex]:
[tex]\[ m_{\text{perpendicular}} = -\frac{1}{-\frac{1}{2}} = 2 \][/tex]
Thus, the slope of the perpendicular line is [tex]\(2\)[/tex].
Step 3: Use the point-slope form to find the equation
We need the equation of the line that passes through the point [tex]\((-5, -1)\)[/tex] with the slope [tex]\(2\)[/tex].
The point-slope form of a line equation is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here [tex]\( (x_1, y_1) = (-5, -1) \)[/tex] and [tex]\( m = 2 \)[/tex]. Substituting these values in:
[tex]\[ y - (-1) = 2(x - (-5)) \][/tex]
Simplifying this equation step-by-step:
[tex]\[ y + 1 = 2(x + 5) \][/tex]
Step 4: Convert to slope-intercept form (y = mx + b)
Distribute the 2 on the right-hand side:
[tex]\[ y + 1 = 2x + 10 \][/tex]
Subtract 1 from both sides to solve for y:
[tex]\[ y = 2x + 9 \][/tex]
So the equation of the line in slope-intercept form that is perpendicular to [tex]\( y = -\frac{1}{2} x - 5 \)[/tex] and passes through the point [tex]\((-5, -1)\)[/tex] is:
[tex]\[ y = 2x + 9 \][/tex]
Here’s the final answer in the requested forms:
Point-Slope Form:
[tex]\[ y - (-1) = 2(x - (-5)) \][/tex]
[tex]\[ y + 1 = 2(x + 5) \][/tex]
Slope-Intercept Form:
[tex]\[ y = 2x + 9 \][/tex]
